I interpret Tolkien's views in a very different way from Richard. Tolkien said Gollum did come incredibly close to repentance, and the only reason he didn't become a better person is because it would have screwed up the story's climax. I will borrow here from Jason Baker's answer to another question.
Did Sméagol legitimately care about Frodo?
Yes, very much. Frodo was the first living thing in a very long time that was nice to him, and he responded to that. He says this explicitly in The Two Towers:
"'But Sméagol said he would be very very good. Nice hobbit! [Frodo] took cruel rope off Sméagol's leg. He speaks nicely to me.'"
- The Two Towers Book IV Chapter 2: "The Passage of the Marshes"
Tolkien goes so far as to say that Frodo may have been able to redeem Sméagol; he says this a few times in his letters, for example in Letter 246 he says:
"If [Sam] had understood better what was going on between Frodo and Gollum, things might have turned out differently in the end. For me perhaps the most tragic moment in the Tale comes in II 323 ff. when Sam fails to note the complete change in Gollum's tone and aspect. 'Nothing, nothing', said Gollum softly. 'Nice master!'. His repentance is blighted and all Frodo's pity is (in a sense) wasted. Shelob's lair became inevitable."
"Sam could hardly have acted differently. (He did reach the point of pity at last (III 221-222) but for the good of Gollum too late.) If he had, what could then have happened? The course of the entry into Mordor and the struggle to reach Mount Doom would have been different, and so would the ending. The interest would have shifted to Gollum, I think, and the battle that would have gone on between his repentance and his new love on one side and the Ring. Though the love would have been strengthened daily it could not have wrested the mastery from the Ring. I think that in some queer twisted and pitiable way Gollum would have tried (not maybe with conscious design) to satisfy both. Certainly at some point not long before the end he would have stolen the Ring or taken it by violence (as he does in the actual Tale). But 'possession' satisfied, I think he would then have sacrificed himself for Frodo's sake and have voluntarily cast himself into the fiery abyss."
"I think that an effect of his partial regeneration by love would have been a clearer vision when he claimed the Ring. He would have perceived the evil of Sauron, and suddenly realized that he could not use the Ring and had not the strength or stature to keep it in Sauron's despite: the only way to keep it and hurt Sauron was to destroy it and himself together – and in a flash he may have seen that this would also be the greatest service to Frodo."
- The Letters of J.R.R. Tolkien 246: To Mrs. Eileen Elgar (Draft). September 1963
In the Dead Marshes, when Sam hears Gollum and Smeagol arguing while Frodo sleeps, Smeagol initially refuses - quite adamantly - to even consider the possibility of hurting Frodo, and - surprisingly - even Gollum concurs:
Not hurt the nice hobbit, of course, no, no.
However, Gollum is more than willing to hurt Sam:
Make the other hobbit, the nasty suspicious hobbit, make him crawl, yes, gollum!
Smeagol has no problem with this; rather than objecting to the idea of hurting Sam, he merely checks to make sure Frodo won't be harmed. Gollum agrees that Frodo won't be hurt, then tries to undermine Smeagol's affection for Frodo. He points out that Frodo is a Baggins, and a Baggins stole the precious; even if Frodo wasn't the thief, he never tried to return the stolen property. Gollum insists that all Bagginses are the enemy. Again, Smeagol resists:
No, not this Baggins!
Gollum only gains ground when he changes the subject a bit, and gets Smeagol thinking about how wonderful it would be to have the precious back. He avoids the issue of how they could get the precious without hurting - probably killing - Frodo, and Smeagol takes the bait without noticing this omission. Gollum gently leads Smeagol along, painting pictures in his head of how great in will be to be reunited with the precious. Finally, he drops the bomb - they should bring Frodo and Sam to Shelob's lair and let her deal with the hobbits. Smeagol snaps out of his trance and recoils in horror - he still refuses to contemplate bringing the "nice hobbit" into harm's way. Frustrated, Gollum drops the subject, saying that perhaps it is too soon to kill off the hobbits; there the conversation ends.
In short, yes, Gollum cared for Frodo - quite deeply, in fact. When he saw Frodo sleeping in Sam's lap on the stairs near Cirith Ungol, he had a change of heart. He saw the beauty and kindness of Frodo, saw the pain the Ring was causing him, and was very sympathetic. He reached out and, in Tolkien's words, "caressed" Frodo's knee. Unfortunately, this made Frodo cry out in his sleep. Sam woke up, assumed that Gollum was up to no good, and chastised him rather harshly.
This hostility from Sam is what destroyed Gollum's chance at redemption.. Gollum recoiled, took a defensive stance, and resolved to lead the hobbits to Shelob, as he had been planning to do up until he had his moment of doubt before the sleeping Frodo's feet.
However, the whole time that Smeagol was flirting with the idea of aligning himself with Frodo, Gollum was fighting it tooth and nail. This delayed any possible alliance, and eventually, it was too late for Gollum, or even Smeagol, to change their stripes. But Smeagol, at least, came very close to joining himself to Frodo and his cause.
We have to remember that, as Jason said, Frodo was the first person to treat Gollum in a remotely kind manner for well over 400 years. Frodo defended Gollum from Sam, from Faramir, from the Rangers, etc - no one had shown such decency to Gollum for as long as he could remember. This was not really wasted on him until Sam yelled at him at precisely the moment that Gollum stood upon the razor's edge, teetering towards repentance and redemption. When Sam shouted at Gollum, it erased any possibility that Gollum would redeem himself.
I close with a final quote from Tolkien's pen:
By temporizing, not fixing the still not wholly corrupt Sméagol-will towards good in the debate in the slag hole, he weakened himself for the final chance when dawning love of Frodo was too easily withered by the jealousy of Sam before Shelob's lair. After that he was lost.
-The Letters of J.R.R. Tolkien 181: To Michael Straight (Draft). January or February 1956
So the debate- which I quoted above- was a very real one. Good Smeagol and bad Gollum were genuinely fighting over what should be done with Frodo and Sam. And in the end, just outside of Shelob's Lair, it was Gollum's jealousy of Samwise that led him to reject his last chance for redemption.
How could Gollum be jealous of Sam's relationship with Frodo, if Gollum himself never loved Frodo? I don't think he could have been jealous of Sam unless he, too, loved their mutual master. In a very real way, it was actually Gollum's love for Frodo that made redemption impossible. He was not able, or not willing, to share Frodo's affection.